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Error Bounds for Gauss-Turán Quadrature Formulae of Analytic Functions

Gradimir V. Milovanović and Miodrag M. Spalević
Mathematics of Computation
Vol. 72, No. 244 (Oct., 2003), pp. 1855-1872
Stable URL: http://www.jstor.org/stable/4100023
Page Count: 18
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Error Bounds for Gauss-Turán Quadrature Formulae of Analytic Functions
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Abstract

We study the kernels of the remainder term $R_{n,s}(f)$ of $Gauss-Tur\acute{a}n$ quadrature formulas $\int_{-1}^{1} f(t)w(t)dt = \sum\limits_{\nu=1}^{n} \sum\limits_{i=0}^{2s} A_{i, \nu}f^{(i)}(\tau_{\nu}) + R_{n,s}(f)\;\;\;\;\;(n \in \mathbb{N}; s \in \mathbb{N}_{0})$ for classes of analytic functions on elliptical contours with foci at ± 1, when the weight w is one of the special Jacobi weights $w^{(\alpha, \beta)}(t) = (1 - t)^{\alpha}(1 + t)^\beta$ ($\alpha = \beta = -1/2$; $\alpha = \beta = 1/2 + s$; $\alpha = -1/2$, $\beta = 1/2 + s$; $\alpha = 1/2 + s$, $\beta = -1/2$). We investigate the location on the contour where the modulus of the kernel attains its maximum value. Some numerical examples are included.

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